A361658 Constant term in the expansion of (1 + x^3 + y^3 + z^3 + 1/(x*y*z))^n.

1, 1, 1, 1, 1, 1, 121, 841, 3361, 10081, 25201, 55441, 194041, 1287001, 7927921, 38438401, 152312161, 516079201, 1627691521, 5745472321, 25999820401, 133086258481, 651284938921, 2860955078521, 11312609403481, 42039298455001, 158864460354601, 658342633033801

Offset

0,7

Links

Table of n, a(n) for n=0..27.

Formula

a(n) = n! * Sum_{k=0..floor(n/6)} 1/(k!^3 * (3*k)! * (n-6*k)!) = Sum_{k=0..floor(n/6)} binomial(n,6*k) * A001421(k).

From Vaclav Kotesovec, Mar 20 2023: (Start)

Recurrence: (n-4)*(n-2)*n^3*a(n) = (6*n^5 - 45*n^4 + 112*n^3 - 123*n^2 + 68*n - 15)*a(n-1) - 3*(n-1)*(5*n^4 - 40*n^3 + 111*n^2 - 132*n + 59)*a(n-2) + 2*(n-2)*(n-1)*(10*n^3 - 75*n^2 + 181*n - 144)*a(n-3) - (n-3)*(n-2)*(n-1)*(15*n^2 - 90*n + 133)*a(n-4) + 3*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 7)*a(n-5) + 1727*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6).

a(n) ~ (1 + 2*sqrt(3))^(n + 3/2) / (4 * 3^(1/4) * Pi^(3/2) * n^(3/2)). (End)

Mathematica

Table[n!*Sum[1/(k!^3 * (3*k)! * (n-6*k)!), {k, 0, n/6}], {n, 0, 30}] (* Vaclav Kotesovec, Mar 20 2023 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\6, 1/(k!^3*(3*k)!*(n-6*k)!));

Crossrefs

Cf. A002426, A361657.

Cf. A001421, A361637.

Sequence in context: A356630 A306452 A238250 * A293588 A365969 A203959

Adjacent sequences: A361655 A361656 A361657 * A361659 A361660 A361661

Keyword

nonn

Author

Seiichi Manyama, Mar 19 2023

Status

approved